Inverse Gaussian Process Research Articles

Fast Pricing and Calculation of Sensitivities of OTM European Options Under Levy Processes

Fast Fourier transform (FFT) method is now a standard calibration engine. However, in many situations, such as pricing of deep out-of-the-money European options, FFT produces large errors. We propose fast and accurate realizations of Integration-Along-Cut method (IAC method), which explicitly control the error of pricing OTM options. For one strike (and OTM options), IAC is significantly faster than FFT-based methods. Even if prices for many strikes are needed, IAC method, together with quadratic interpolation, successfully competes with CONV method developed by Lord et. al., the COS method suggested by Fang and Oosterlee, the saddlepoint method suggested by Carr and Madan, and the refined and enhanced versions of FFT recently suggested by M.~Boyarchenko and Levendorskii. For calculations of sensitivities, a relative advantage of IAC is even greater. The method is applicable to a wide class of Levy-driven models, KoBoL processes (a.k.a. CGMY model) of finite variation, Variance Gamma processes and Normal Inverse Gaussian processes including.

Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Barrier options under wide classes of L\'evy processes with exponential jump densities, including Variance Gamma model, KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes, are studied. The leading term of asymptotics of the option price and the leading term of asymptotics in Carr's randomization approximation to the price are calculated, as the price of the underlying approaches the barrier. We prove that the order of asymptotics is the same in both cases, and the asymptotic coefficient in the asymptotic formula for Carr's randomization approximation converges to the asymptotic coefficient for the price, as the number of time steps N → ∞, and justify Richardson extrapolation of arbitrary order. Similar results are derived for sensitivities and the leading terms of their asymptotics in Carr's randomization approximation. The convergence of prices and sensitivities is proved in appropriate weighted H¨older spaces.

Prices of Barrier and First-Touch Digital Options in Levy-Driven Models, Near Barrier

We calculate the leading term of asymptotics of the prices of barrier options and first touch digitals near the barrier for wide classes of Levy processes with exponential jump densities, including Variance Gamma model, KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. In many cases, we calculate the second term of asymptotics as well.

Approximation for the Normal Inverse Gaussian Process Using Random Sums

We approximate the normal inverse Gaussian (NIG) process with random summations. The random sum we introduce is a random walk subordinated to the first passage time of another independent random walk; the model is interpreted as an internal mechanism at small scale that generates the NIG process. The main result is a functional limit theorem of weak convergence in the Skorohod topology.

A THEORETICAL ANALYSIS OF MULTISCALE ENTROPY UNDER THE INVERSE GAUSSIAN DISTRIBUTION

Multiscale entropy (MSE) discloses the intrinsic multiple scales in the complexity of physical and physiological signals, which are usually featured by heavy-tailed distributions. Most of these research results are pure experimental search, till Costa et al. made the first attempt to the theoretical basis of MSE. However, the analysis only supports the Gaussian distribution [Phys. Rev. E71, 021906 (2005)]. In this paper, we present the theoretical basis of MSE under the inverse Gaussian distribution, which is a typical model for physiological, physical and financial data sets. The analysis is applicable to uncorrelated inverse Gaussian process and 1/f noise with the multivariate inverse Gaussian distribution, providing a reliable foundation for potential applications of MSE to explore complex physical and physiological time series.

Pricing of hourly exercisable electricity swing options using different price processes

In this paper fair values of hourly exercisable swing options written as the underlying on the EEX spot price are calculated using three different two-factor models: a regime-switching AR process, a jump-diffusion process with Bernoulli jump terms and a normal inverse Gaussian process. An efficient least squares Monte Carlo algorithm is introduced and applied to swing options with up to 5,000 exercise rights. Finally, the three models are compared with a focus on their ability to reproduce the characteristics of the EEX spot prices and the swing option values resulting for different numbers of exercise rights.

Semiparametric inference on a class of Wiener processes

Abstract. This article studies the estimation of a nonhomogeneous Wiener process model for degradation data. A pseudo-likelihood method is proposed to estimate the unknown parameters. An attractive algorithm is established to compute the estimator under this pseudo-likelihood formulation. We establish the asymptotic properties of the estimator, including consistency, convergence rate and asymptotic distribution. Random effects can be incorporated into the model to represent the heterogeneity of degradation paths by letting the mean function be random. The Wiener process model is extended naturally to a normal inverse Gaussian process model and similar pseudo-likelihood inference is developed. A score test is used to test the presence of the random effects. Simulation studies are conducted to validate the method and we apply our method to a real data set in the area of health structure monitoring.

Valuation of Continuously Monitored Double Barrier Options and Related Securities

In this article we apply Carr's randomization approximation and the operator form of the Wiener-Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain contingent claims with first-touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options.Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Levy-driven models including Variance Gamma processes, Normal Inverse Gaussian processes and KoBoL processes (a.k.a. the CGMY model). At the same time, our work gives new insight into the known explicit formulas obtained by other authors in the setting of the Black-Scholes model. The operator form of the Wiener-Hopf method is generalized for wide classes of processes including the important class of Variance Gamma processes.Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock-out double barrier put/call options as well as double-no-touch options.

On option pricing under a completely random measure via a generalized Esscher transform

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771–1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416–440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.

Pricing Options Under a Generalized Markov-Modulated Jump-Diffusion Model

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.

Comparison of semimartingales and Lévy processes

In this paper, we derive comparison results for terminal values of d-dimensional special semimartingales and also for finite-dimensional distributions of multivariate Lévy processes. The comparison is with respect to nondecreasing, (increasing) convex, (increasing) directionally convex and (increasing) supermodular functions. We use three different approaches. In the first approach, we give sufficient conditions on the local predictable characteristics that imply ordering of terminal values of semimartingales. This generalizes some recent convex comparison results of exponential models in [Math. Finance 8 (1998) 93–126, Finance Stoch. 4 (2000) 209–222, Proc. Steklov Inst. Math. 237 (2002) 73–113, Finance Stoch. 10 (2006) 222–249]. In the second part, we give comparison results for finite-dimensional distributions of Lévy processes with infinite Lévy measure. In the first step, we derive a comparison result for Markov processes based on a monotone separating transition kernel. By a coupling argument, we get an application to the comparison of compound Poisson processes. These comparisons are then extended by an approximation argument to the ordering of Lévy processes with infinite Lévy measure. The third approach is based on mixing representations which are known for several relevant distribution classes. We discuss this approach in detail for the comparison of generalized hyperbolic distributions and for normal inverse Gaussian processes.

Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes

Lévy processes can be used to model asset return's distributions. Monte Carlo methods must frequently be used to value path dependent options in these models, but Monte Carlo methods can be prone to considerable simulation bias when valuing options with continuous reset conditions. This paper shows how to correct for this bias for a range of options by generating a sample from the extremes distribution of the Lévy process on subintervals. The method uses variance‐gamma and normal inverse Gaussian processes. The method gives considerable reductions in bias, so that it becomes feasible to apply variance reduction methods. The method seems to be a very fruitful approach in a framework in which many options do not have analytical solutions.

On The Expected Discounted Penalty function for Lévy Risk Processes

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes. In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.

PRICING OF FIRST TOUCH DIGITALS UNDER NORMAL INVERSE GAUSSIAN PROCESSES

We calculate prices of first touch digitals under normal inverse Gaussian (NIG) processes, and compare them to prices in the Brownian model and double exponential jump-diffusion model. Numerical results are produced to show that for typical parameters values, the relative error of the Brownian motion approximation to NIG price can be 2–3 dozen percent if the spot price is at the distance 0.05–0.2 from the barrier (normalized to one). A similar effect is observed for approximations by the double exponential jump-diffusion model, if the jump component of the approximation is significant. We show that two jump-diffusion processes can give approximately the same results for European options but essentially different results for first touch digitals and barrier options. A fast approximate pricing formula under NIG is derived.

A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

This paper gives a tree‐based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American‐type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path‐dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.

GARCH-type volatility models based on Brownian inverse Gaussian intra-day return processes

GARCH models are useful to estimate daily volatility in financial return series. When intra-day return data are available realized volatility may be used for the same purpose. We formulate a new model that is a hybrid between a traditional GARCH model and a stochastic volatility model. This model postulates that over each trading day anew the intra-day return process follows a Brownian motion with drift and volatility that are inherited from the previous day in typical GARCH fashion but are also subject to random inverse Gaussian (IG) distributed news noise impacts that arrive after market closure on the previous day. We call it the BIG–GARCH model and derive the likelihood function needed to fit it. It turns out that this uses the daily returns and realized volatilities as sufficient statistics. To make this possible we introduce a number of new distributions related to the IG-distribution and also derive diagnostics that may be used to check the quality of fit. The new model produces two volatility measures, called the expected volatility and the actual volatility. We show that the latter is close to the realized volatility. The model is illustrated by fitting it to IBM daily and intra-day returns.

An Examination on the Roles of Diffusions and Stochastic Volatility in the Exponential Levy Jumps Models

Recent studies have shown that stochastic volatility in a continuous-time framework provides an excellent fit for financial asset returns when combined with finite-activity Merton's type compound Poisson Jump-diffusion models. However, we demonstrate that stochastic volatility does not play a central role when incorporated with infinite-activity Levy type pure jump models such as variance-gamma and normal inverse Gaussian processes to model high and low frequency historical time-series SP500 index returns. In addition, whether sources of stochastic volatility are diffusions or jumps are not relevant to improve the overall empirical fits of returns. Nevertheless, stochastic diffusion volatility with infinite-activity Levy jumps processes considerably reduces SP500 index call option in-sample and out-of-sample pricing errors of long-term ATM and OTM options, which contributed to a substantial improvement of pricing performances of SVJ and EVGSV models, compared to constant volatility Levy-type pure jumps models and/or stochastic volatility model without jumps. Interestingly, unlike asset returns, whether pure Levy jumps specifications are finite or infinite activity is not an important factor to enhance option pricing model performances once stochastic volatility is incorporated. Option prices are computed via improved Fast Fourier Transform algorithm using characteristic functions to match arbitrary log-strike grids with equal intervals with each moneyness and maturity of actual market option prices considered in this paper.

Path Dependant Option Pricing under Levy Processes

A model is developed that can price path dependent options when the underlying process is an exponential Levy process with closed form conditional characteristic function. The model is an extension of a recent quadrature option pricing model so that it can be applied with the use of Fourier and Fast Fourier transforms. Thus the model possesses nice features of both transform and quadrature option pricing techniques since it can be applied for a very general set of underlying Levy processes and can handle exotic path dependent features. The model is applied to European and Bermudan options for geometric Brownian motion, a jump-diffusion process, a variance gamma process and a normal inverse Gaussian process. However it must be noted that the model can also price other path dependent exotic options such as lookback and Asian options.

Risk Theory with the Generalized Inverse Gaussian Lévy Process

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

Risk Theory with the Generalized Inverse Gaussian Lévy Process

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.